The rational homology of real toric manifolds Alexander I. Suciu Toric manifolds. In a seminal paper [7] that appeared some twenty years ago, Michael Davis and Tadeusz Januszkiewicz introduced a topological version of smooth toric varieties, and showed that many properties previously discovered by means of algebro-geometric techniques are, in fact, topological in nature. Let P be an n -dimensional simple polytope with facets F 1 , . . . , F m , and let χ be an integral n × m matrix such that, for each vertex v = F i 1 ∩ · · · ∩ F i n , the minor of columns i 1 , . . . , i n has determinant ± 1. To such data, there is associated a 2 n -dimensional toric manifold, M P ( χ ) = T n × P/ ∼ , where ( t, p ) ∼ ( u, q ) if p = q , and tu − 1 belongs to the image under χ : T m → T n of the coordinate subtorus corresponding to the smallest face of P containing q in its interior. Here is an alternate description, using the moment-angle complex construction (see for instance [10] and references therein). Given a simplicial complex K on vertex set [ n ] = { 1 , . . . , n } , and a pair of spaces ( X, A ), let Z K ( X, A ) be the subspace of the cartesian product X × n , defined as the union � σ ∈ K ( X, A ) σ , where ( X, A ) σ is the set of points for which the i -th coordinate belongs to A , whenever i / ∈ σ . It turns out that the quasi-toric manifold M P ( χ ) is obtained from the moment angle manifold Z K ( D 2 , S 1 ), where K is the dual to ∂P , by taking the quotient by the relevant free action of the torus T m − n = ker( χ ). Real toric manifolds. An analogous theory works for real quasi-toric manifolds, also known as small covers. Given a homomorphism χ : Z m → Z n 2 satisfying a 2 minors condition as above, the resulting n -dimensional manifold, N P ( χ ), is the quotient of the real moment angle manifold Z K ( D 1 , S 0 ) by a free action of the group Z m − n = ker( χ ). The manifold N P ( χ ) comes equipped with an action of Z n 2 ; 2 the associated Borel construction is homotopy equivalent to Z K ( RP ∞ , ∗ ). If X is a smooth, projective toric variety, then X ( C ) = M P ( χ ), for some simple polytope P and characteristic matrix χ , and X ( R ) = N P ( χ mod 2 Z ). Not all toric manifolds arise in this manner. For instance, M = CP 2 ♯ CP 2 is a toric manifold over the square, but it does not admit any (almost) complex structure; thus, M �∼ = X ( C ). The same goes for real toric manifolds. For instance, take P to be the dodec- ahedron, and use one of the characteristic matrices χ listed in [12]. Then, by a theorem of Andreev [1], the small cover N P ( χ ) is a hyperbolic 3-manifold; thus, by a theorem of Delaunay [8], N P ( χ ) �∼ = X ( R ). The Betti numbers of real toric manifolds. In [7], Davis and Januszkiewicz showed that the sequence of mod 2 Betti numbers of N P ( χ ) coincides with the h -vector of P . In joint work with Alvise Trevisan [18], we compute the rational cohomology groups (together with their cup-product structure) for real, quasi- toric manifolds. It turns out that the rational Betti numbers are much more subtle, depending also on the characteristic matrix χ . 1
� � More precisely, for each subset S ⊆ [ n ], let χ S = � i ∈ S χ i , where χ i is the i -th row of χ , and let K χ,S be the induced subcomplex of K on the set of vertices j ∈ [ m ] for which the j -th entry of χ S is non-zero. Then � dim � (*) dim H q ( N P ( χ ) , Q ) = H q − 1 ( K χ,S , Q ) . S ⊆ [ n ] The proof of formula (*), given in [18], relies on two fibrations relating the real toric manifold N P ( χ ) to some of the aforementioned moment-angle complexes, Z m − n 2 Z K ( D 1 , S 0 ) � N P ( χ ) � Z K ( RP ∞ , ∗ ) . Z n 2 The proof entails a detailed analysis of homology in rank 1 local systems on the space Z K ( RP ∞ , ∗ ), exploiting at some point the stable splitting of moment-angle complexes due to Bahri, Bendersky, Cohen, and Gitler [2]. Some of the details of the proof appear in Trevisan’s Ph.D. thesis [19]. As an easy application of formula (*), one can readily recover a result of Nakayama and Nishimura [14]: A real, n -dimensional toric manifold N P ( χ ) is orientable if and only if there is a subset S ⊆ [ n ] such that K χ,S = K . The Hessenberg varieties. A classical construction associates to each Weyl group W a smooth, complex projective toric variety T W , whose fan corresponds to the reflecting hyperplanes of W and its weight lattice. In the case when W is the symmetric group S n , the manifold T n = T S n is the well-known Hessenberg variety, see [9]. Moreover, T n is isomorphic to the De Concini–Procesi wonderful model Y G , where G is the maximal building set for the Boolean arrangement in CP n − 1 . Thus, T n can be obtained by iterated blow- ups: first blow up CP n − 1 at the n coordinate points, then blow up along the proper � n � transforms of the coordinate lines, etc. 2 The real locus, T n ( R ), is a smooth, real toric variety of dimension n − 1; its rational cohomology was recently computed by Henderson [13], who showed that � n � dim H i ( T n ( R ) , Q ) = A 2 i , 2 i where A 2 i is the Euler secant number, defined as the coefficient of x 2 i / (2 i )! in the Maclaurin expansion of sec( x ). As announced in [17], we can recover this computation, using formula (*). To start with, note that the ( n − 1)-dimensional polytope associated to T n ( R ) is the permutahedron P n . Its vertices are obtained by permuting the coordinates of the vector (1 , . . . , n ) ∈ R n , while its facets are indexed by the non-empty, proper subsets Q ⊂ [ n ]. The characteristic matrix χ = ( χ Q ) for T n ( R ) can be described 2
as follows: χ i is the i -th standard basis vector of R n − 1 for 1 ≤ i < n , while χ n = � i<n χ i and χ Q = � i ∈ Q χ i . The simplicial complex K n dual to ∂P n is the barycentric subdivision of the boundary of the ( n − 1)-simplex. Given a subset S ⊂ [ n − 1], the induced sub- complex ( K n ) χ,S depends only on the cardinality r = | S | ; denote any one of these � n − 1 � subcomplexes by K n,r . It turns out that K n,r is the order complex associ- r ated to a rank-selected poset of a certain subposet of the Boolean lattice B n . A result of Bj¨ orner and Wachs [5] insures that such simplicial complexes are Cohen– Macaulay, and thus have the homotopy type of a wedge of spheres (of a fixed dimension); in fact, K n, 2 r − 1 ≃ K n, 2 r ≃ � A 2 r S r − 1 . Hence, � dim � dim H i ( T n ( R ) , Q ) = H i − 1 (( K n ) χ,S , Q ) S ⊆ [ n − 1] � n − 1 � n − 1 � dim � = H i − 1 ( K n,r , Q ) r r =1 �� n − 1 � � n − 1 �� � n � = + A 2 i = A 2 i . 2 i − 1 2 i 2 i Recently, Choi and Park [6] have extended this computation to a much wider class of real toric manifolds. Given a finite simple graph Γ, let B (Γ) be the building set obtained from the connected induced subgraphs of Γ, and let P B (Γ) be the corresponding graph associahedron. Using formula (*), these authors compute the Betti numbers of the smooth, real toric variety X Γ ( R ) defined by P B (Γ) . When Γ = K n is a complete graph, X K n = T n , and one recovers the above calculation. The formality question. A finite-type CW-complex X is said to be formal if its Sullivan minimal model is quasi-isomorphic to the rational cohomology ring of X , endowed with the 0 differential. Under a nilpotency assumption, this means that H ∗ ( X, Q ) determines the rational homotopy type of X . As shown by Notbohm and Ray [15], if X is formal, then Z K ( X, ∗ ) is formal; in particular, Z K ( S 1 , ∗ ) and Z K ( CP ∞ , ∗ ) are always formal. More generally, as shown by F´ elix and Tanr´ e [11], if both X and A are formal, and the inclusion A ֒ → X induces a surjection in rational cohomology, then Z K ( X, A ) is formal. On the other hand, as sketched in [4], and proved with full details in [10], the spaces Z K ( D 2 , S 1 ) can have non-trivial triple Massey products, and thus are not always formal. In fact, as shown in [10], there exist polytopes P and dual triangulations K = K ∂P for which the moment-angle manifold Z K ( D 2 , S 1 ) is not formal. Using these results, as well as a construction from [3], we can exhibit real moment-angle manifolds Z L ( D 1 , S 0 ) that are not formal. In view of this discussion, the following natural question arises: are toric mani- folds formal? Of course, smooth (complex) toric varieties are formal, by a classical result of Deligne, Griffith, Morgan, and Sullivan. More generally, Panov and Ray showed in [16] that all toric manifolds are formal. So we are left with the question whether real toric manifolds are always formal. 3
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